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The stable marriage problem an example

A 5/3 approximation algorithm for a hard case of stable marriage Rob Irving Computing Science Department University of Glasgow (joint work with David Manlove ). The stable marriage problem an example. m 1 : w 2 w 4 w 1 w 3 w 1 : m 2 m 1 m 4

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The stable marriage problem an example

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  1. A 5/3 approximation algorithm for a hard case of stable marriageRob IrvingComputing Science DepartmentUniversity of Glasgow(joint work with David Manlove) LAW February 2007

  2. The stable marriage problem an example m1: w2 w4 w1 w3 w1: m2 m1 m4 m2: w3 w1 w4 w2: m4 m3 m1 m3: w2 w3 w3: m1 m4 m3 m2 m4: w4 w1 w3 w2 w4: m2 m1 m4 men’s preferences women’s preferences An example matching - {(m1, w1), (m2, w3), (m3, w2), (m4, w4)} It's an unstable matching - (m1, w4)is a blocking pair LAW February 2007

  3. The stable marriage problem an example m1: w2 w4 w1 w3 w1: m2 m1 m4 m2: w3 w1 w4 w2: m4 m3 m1 m3: w2 w3 w3: m1 m4 m3 m2 m4: w4 w1 w3 w2 w4: m2 m1 m4 men’s preferences women’s preferences {(m1, w4), (m2, w1), (m3, w2), (m4, w3)}- a stable matching LAW February 2007

  4. Stable marriage facts • There is always at least one stable matching • The Gale-Shapley algorithm finds such a matching [Gale & Shapley 1962] • There may be many stable matchings • exponentially many in some cases [Knuth 1976] • All stable matchings have the same size and involve the same people [Gale & Sotomayor 1985] • The version of stable marriage described here is usually called "Stable Marriage with Incomplete Lists" (SMI) LAW February 2007

  5. The Hospitals / Residents probleman example r1: h4 h2 h1 h1: 2: r2 r7 r1 r5 r4 r6 r3 r2: h3 h1 h4 h2: 2: r8r1 r3 r6 r3: h2 h3 h1 h3: 2: r3 r5 r2 r6 r8 r7 r4 r4: h3 h4 h1 h4: 3: r5 r8 r2 r4 r7r1 r5: h1 h4 h3 r6: h2 h3 h1 r7: h4 h1 h3 r8: h3 h4 h2 residents’ preferences hospitals’ quotas and preferences An example matching - {(r1,h2),(r2,h3),(r3,h3),(r4,h4),(r5,h4),(r6,h2),(r7,h1),(r8,h4)} It's unstable - (r3,h2)is a blocking pair so is(r5,h1) LAW February 2007

  6. The Hospitals / Residents problem an example r1: h4 h2 h1 h1: 2: r2 r7 r1 r5 r4 r6 r3 r2: h3 h1 h4 h2: 2: r8r1 r3 r6 r3: h2 h3 h1 h3: 2: r3 r5 r2 r6 r8 r7 r4 r4: h3 h4 h1 h4: 3: r5 r8 r2 r4 r7r1 r5: h1 h4 h3 r6: h2 h3 h1 r7: h4 h1 h3 r8: h3 h4 h2 residents’ preferences hospitals’ capacities and preferences A stable matching {(r1,h2),(r2,h3),(r3,h2),(r4,h4),(r5,h1),(r6,h3),(r7,h4),(r8,h4)} LAW February 2007

  7. Stable matching in practice • The National Resident Matching Program (NRMP) • a scheme that each year matches over 30,000 graduating medical students (residents) to hospitals in the US • Since the early 1950s, the NRMP has used essentially the Gale-Shapley algorithm [Roth 1984] • Owes its success to the stability of its matching • Similar matching schemes in other countries and other contexts • including the ‘SFAS’ scheme in Scotland [Irving 1998] LAW February 2007

  8. Ties in the preference lists • Stable Marriage with Ties and Incomplete Lists (SMTI) • Hospitals / Residents with Ties (HRT) • Each preference list can contain one or more ties • arguably a more realistic model in practical applications • large hospitals cannot reasonably differentiate among all of their many applicants • Now, a pair blocks a matching only if both members of the pair would be better off if matched together • sometimes called weak stability • To find a stable matching, break the ties arbitrarily and apply the Gale-Shapley algorithm • But now – stable matchings can have different sizes LAW February 2007

  9. SMTI - an illustration m1: w3 (w1 w2) w4 w1: m1 m2 m3 m4 m2: w1 w3 w2: m1 m4 m3: w3 w1 w3: m4 m3 m1 m2 m4: w1 (w3 w4) w2 w4: m4 m1 men’s preferences women’s preferences A stable matching of size 4: {(m1, w2), (m2, w1), (m3, w3), (m4, w4)} A stable matching of size 2: {(m1, w1), (m4, w3)} LAW February 2007

  10. A hard stable matching problem • For Stable Marriage (and therefore Hospitals Residents) with incomplete lists and ties • It's NP-hard to find a maximum cardinality stable matching • even if • all the ties are on one side, and • a tie occurs only at the end of a list, and • all ties are of length 2 • [Iwama, Manlove et al 1999] • [Manlove, Irving et al 2002] • Question: what about polynomial-time approximation? • Can we decide on a 'good' way to break the ties? LAW February 2007

  11. Approximating maximum cardinality SMTI • Trivially 2-approximable • max cardinality ≤ 2 * min cardinality • But APX-complete • [HIIMMMS 2003] • And not approximable within 21/19 (unless P = NP) • 13/7-approximable if all ties are of length 2 • 8/5-approximable if all ties are of length 2 and on just one side • 10/7 approximable (randomised) if, in addition, ≤ 1 tie per list • [Halldórsson et al 2003] • 15/8 approximable in the general case • [Iwama et al, SODA 2007] LAW February 2007

  12. A case of practical interest • 'Special' HRT (SHRT) • residents' preference lists are strict • hospitals' lists can contain one tie, of arbitrary length, at the end • reflects the situation where hospitals are asked to rank as many candidates as they reasonably can, then place all others in a single tie at the end of their list • 'Special' SMTI (SSMTI) • men's preference lists are strict • women's lists can contain one tie, of arbitrary length, at the end • MAX-SSMTI is NP-hard (and so MAX-SHRT is too) LAW February 2007

  13. Approximating Max-SSMTI and MAX-SHRT • There is a polynomial-time approximation algorithm for maximum cardinality SSMTI ( and for SHRT) with an approximation guarantee of 5/3. • Algorithm is quite different from the approach used by previous approximation algorithms for variants of MAX-SMTI • Consists of three phases: • Phase 1 : apply a tailored version of the Gale-Shapley algorithm • Phase 2 : find a maximum cardinality matching of selected subsets • Phase 3 : break the ties according to a particular rule, then apply standard Gale-Shapley LAW February 2007

  14. Algorithm SSMTI-APPROX: Phase 1 assign each person to be free; /* free = not engaged */ while (some woman w is free) and (w has a non-empty list) and (w has an untied man m at the head of her list) { w proposes, and becomes engaged to m; /* m can never have a 'worse' mate than w; in particular, m is matched in every stable matching */ for each successor w' of w on m's list { if w' is engaged to m break the engagement; delete the pair (m,w') from the preference lists; } } Fact: engaged men are matched in every stable matching LAW February 2007

  15. Algorithm SSMTI-APPROX: Phase 1 m1 : w6 w1 w2 w10 w5 w8 w1 : m2 m3 (m1 m6 m9) m2 : w7 w10 w3 w1 w6 w2 : m1 (m3 m5 m7) m3 : w6 w2 w8 w9 w1 w3 : m2 (m4 m6 m8) m4 : w7 w9 w8 w3 w4 w5 w4 : m4 m9 (m5 m8 m10) m5 : w2 w6 w4 w5 w10 w5 : m4 m1 m5 m7 m6 : w1 w3 w6 : m2 (m1 m3 m5) m7 : w2 w5 w7 : (m2 m4 m10) m8 : w3 w4 w8 : m1 (m3 m4) m9 : w1 w4 w9 : (m3 m4) m10: w7 w10 w4 w10: m5 (m1 m2 m10) men’s preferences women’s preferences LAW February 2007

  16. Algorithm SSMTI-APPROX: Phase 1 outcome m1: w6 w1 w2w1: m3 (m1 m6 m9) m2: w7 w10 w3w2: m1 (m3 m5 m7) m3: w6 w2 w8 w9 w1w3: m2 (m4 m6 m8) m4: w7 w9 w8 w3 w4w4: m4 m9 (m5 m8 m10) m5: w2 w6 w4 w5w5: m5 m7 m6 : w1 w3 w6 : (m1 m3 m5) m7 : w2 w5 w7 : (m2 m4 m10) m8 : w3 w4 w8 : (m3 m4) m9 : w1 w4 w9 : (m3 m4) m10: w7 w10 w4 w10: (m2 m10) red people are engaged, black people are free LAW February 2007

  17. Algorithm SSMTI-APPROX: Phase 2 Find a maximum cardinality matching F of free men and women; for each pair (m,w)  F promote m from the tie to the head of w's list; reactivate phase 1; /* essentially extends the sets of engaged men and women */ LAW February 2007

  18. Algorithm SSMTI-APPROX: Phase 2 m1: w6 w1 w2w1: m3 (m1 m6 m9) m2: w7 w10 w3w2: m1 (m3 m5 m7) m3: w6 w2 w8 w9 w1w3: m2 (m4 m6 m8) m4: w7 w9 w8 w3 w4w4: m4 m9 (m5 m8 m10) m5: w2 w6 w4 w5w5: m5 m7 m6 : w1 w3 w6 : (m1 m3 m5) m7 : w2 w5 w7 : (m2 m4 m10) m8 : w3 w4 w8 : (m3 m4) m9 : w1 w4 w9 : (m3 m4) m10: w7 w10 w4 w10: (m2 m10) red people are engaged, black people are free LAW February 2007

  19. Algorithm SSMTI-APPROX: Phase 2 m1: w6 w1 w2w1: m3 (m1 m6 m9) m2: w7 w10 w3w2: m1 (m3 m5 m7) m3: w6 w2 w8 w9 w1w3: m2 (m4 m6 m8) m4: w7 w9 w8 w3 w4w4: m4 m9 (m5 m8 m10) m5: w2 w6 w4 w5w5: m5 m7 m6 : w1 w3 w6 : (m1 m3 m5) m7 : w2 w5 w7 : (m2 m4 m10) m8 : w3 w4 w8 : (m3 m4) m9 : w1 w4 w9 : (m3 m4) m10: w7 w10 w4 w10: m10 m2 red people are engaged, black people are free LAW February 2007

  20. Algorithm SSMTI-APPROX: Phase 2 outcome m1: w6 w1 w2w1: m3 (m1 m6 m9) m2: w7 w10 w3w2: m1 (m3 m5 m7) m3: w6 w2 w8 w9 w1w3: m2 (m4 m6 m8) m4: w7 w9 w8 w3 w4w4: m4 m9 (m5 m8) m5: w2 w6 w4 w5w5: m5 m7 m6 : w1 w3 w6 : (m1 m3 m5) m7 : w2 w5 w7 : (m2 m4 m10) m8 : w3 w4 w8 : (m3 m4) m9 : w1 w4 w9 : (m3 m4) m10: w7 w10w10: m10 m2 red people are engaged, black people are free LAW February 2007

  21. Algorithm SSMTI-APPROX: Phase 3 • for each woman w • break the tie (if any) in w's list, • placing the free men ahead of the engaged men; • reactivate phase 1; /* now just classical Gale-Shapley */ • return the set of engaged pairs; • This is obviously a stable matching • All we have done is decide how to break the ties and then apply Gale-Shapley LAW February 2007

  22. Algorithm SSMTI-APPROX: Phase 3 m1: w6 w1 w2w1: m3 (m1m6 m9) m2: w7 w10 w3w2: m1 (m3 m5m7) m3: w6 w2 w8 w9 w1w3: m2 (m4m6 m8) m4: w7 w9 w8 w3 w4w4: m4 m9 (m5m8) m5: w2 w6 w4 w5w5: m5 m7 m6 : w1 w3 w6 : (m1 m3 m5) m7 : w2 w5 w7 : (m2 m4 m10) m8 : w3 w4 w8 : (m3 m4) m9 : w1 w4 w9 : (m3 m4) m10: w7 w10w10: m10 m2 LAW February 2007

  23. Algorithm SSMTI-APPROX: Phase 3 m1: w6 w1 w2w1: m3 m6 m9 m1 m2: w7 w10 w3w2: m1 m7 m3 m5 m3: w6 w2 w8 w9 w1w3: m2 m6 m8 m4 m4: w7 w9 w8 w3 w4w4: m4 m9 m8 m5 m5: w2 w6 w4 w5w5: m5 m7 m6 : w1 w3 w6 : m1 m3 m5 m7 : w2 w5 w7 : m2 m4 m10 m8 : w3 w4 w8 : m3 m4 m9 : w1 w4 w9 : m3 m4 m10: w7 w10w10: m10 m2 LAW February 2007

  24. Algorithm SSMTI-APPROX: Phase 3 m1: w6w1: m6 m9 m2: w7w2: m7 m3 m5 m3: w6 w2 w8w3: m8 m4: w7 w9w4: m9 m5 m5: w2 w6 w4 w5w5: m5 m6: w1w6: m1 m3 m5 m7: w2w7: m2 m4 m10 m8: w3w8: m3 m9: w1 w4w9: m4 m10: w7 w10w10: m10 A stable matching of size 10 LAW February 2007

  25. Visualising the performance guarantee Extra men are matched in M but not in A A Engaged men Extra women must also be matched in A Extra women M Extra men M = maximum cardinality stable matching A = stable matching found by algorithm SSMTI-APPROX Free men A Women Men LAW February 2007

  26. Visualising the performance guarantee These women must be matched in A M s s A Engaged men x A s Extra women M r-s s Extra men M = maximum cardinality stable matching A = stable matching found by algorithm SSMTI-APPROX r-s Free men A r-s Women Men LAW February 2007

  27. Visualising the performance guarantee These women must be matched in A M s s A • Inequalities: • |A| ≥ |M| - r • |A| ≥ |M| - x (a consequence of Phase 2) • |A| ≥ x + r – s • |A| ≥ r + s • Algebra gives |M| ≤ 5|A|/3 x A s Extra women M r-s s Extra men r-s A r-s Women Men LAW February 2007

  28. Further observations • The algorithm extends in a (fairly) straightforward way to SHRT • with the same performance guarantee of 5/3 • We have no example where the 5/3 bound is realised • The worst case we have is 8/5 • so maybe a sharper analysis could improve the bound • Open question • Can these or similar techniques give good approximation algorithms for more general cases? • e.g. when men also can have a tie at the end of their list LAW February 2007

  29. A bad case for Algorithm SSMTI-APPROX m1 : w5 w2 w1 : m3 (m8 m5) m2 : w6 w3 w2 : m1 m6 m3 : w5 w7 w8 w1 w3 : m2 m7 m4 : w6 w8 w7 w4 w4 : m4 m8 m5 : w1 w5 : (m3 m1) m6 : w2 w6 : (m4 m2) m7 : w3 w7 : (m4 m3) m8 : w1 w4 w8 : (m3 m4) Algorithm can return the red matching – size 5 The blue matching – size 8 – is also stable LAW February 2007

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